
where \(\varepsilon = 0.1\) , \(m = 3\) , and \(n = 2\) . Our numerical simulations, shown in fig:SurfaceTension, are able to reproduce the key morphological features of the Saffman–Taylor instability. For each value of \(\sigma \)  considered, the interface is unstable, and the sinusoidal perturbations in the initial condition (REF ) grow into fingers. As \(\sigma \)  is decreased, the interface becomes more unstable and the number of fingers that develop over the duration of a simulation increases due to tip-splitting occurring (see \((1)\)  in fig:SurfaceTension\((b)\)  for example). Additionally, our simulations are able to produce so called `shielding' behaviour, where neighbouring fingers can block one another off, which in turn results in fingers retracting (denoted by \((2)\)  in fig:SurfaceTension\((c)\) ). This behaviour is known to occur experimentally (see Figure 1 in [1]} for example). Finally, when surface tension is sufficiently small, `feathering' can occur, when a finger does not strictly tip-split, but instead develops ripples along one of its sides as it expands (denoted by \((3)\)  in fig:SurfaceTension\((d)\) ). Again this behaviour has been observed in experiments (see Figure 3 in [2]}).
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